#!/usr/bin/env julia

# Jacek Poziemski 272389

"""
    f(x::Float64)::Float64

Calculate `sin(x) + cos(3x)`.

# Arguments
- `x::Float64`: function argument
"""
f(x::Float64)::Float64 = sin(x) + cos(Float64(3) * x)

"""
    fderivative(x::Float64)::Float64

Calculate the derivative of `sin(x) + cos(3x)`, `cos(x) - 3sin(3x)`.

# Arguments
- `x::Float64`: function argument
"""
fderivative(x::Float64)::Float64 = cos(x) - Float64(3) * sin(Float64(3) * x)

"""
    derivative(f::Function, x0::Float64, h::Float64)::Float64

Calculate an approximation of the derivative of `f` at `x0`,
using `h` in the following formula: `(f(x0 + h) - f(x0)) / h`.

# Arguments
- `f::Function`: function to calculate the derivative of
- `x0::Float64`: function argument
- `h::Float64`: value used in the approximation formula
"""
derivative(f::Function, x0::Float64, h::Float64)::Float64 = (f(x0 + h) - f(x0)) / h

x::Float64 = one(Float64)

for n::Int in 0:54
    h::Float64 = 2.0^(-n)
    d::Float64 = derivative(f, x, h)
    e::Float64 = abs(fderivative(x) - d)
    println("n: $n; h: $h; h + 1: $(h + 1); derivative: $d; error: $e")
end

println("exact value: $(fderivative(1.0))")